Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory

TL;DR

This paper generalizes point processes associated with fermionic gases, random matrices, and zeta zeros to arbitrary dimensions, revealing their hyperuniformity, repulsive nature, and connections to sphere packings.

Contribution

It provides exact analytical formulas for d-dimensional point processes, extending known 1D cases, and explores their properties and implications in higher dimensions.

Findings

Point processes are hyperuniform in all dimensions.

Correlation functions show repulsive behavior and a growing effective hard-core diameter.

Large cavities behave like Poisson processes in dimension d+1.

Abstract

It is well known that one can map certain properties of random matrices, fermionic gases, and zeros of the Riemann zeta function to a unique point process on the real line. Here we analytically provide exact generalizations of such a point process in d-dimensional Euclidean space for any d, which are special cases of determinantal processes. In particular, we obtain the n-particle correlation functions for any n, which completely specify the point processes. We also demonstrate that spin-polarized fermionic systems have these same n-particle correlation functions in each dimension. The point processes for any d are shown to be hyperuniform. The latter result implies that the pair correlation function tends to unity for large pair distances with a decay rate that is controlled by the power law r^[-(d+1)]. We graphically display one- and two-dimensional realizations of the point processes in order to vividly reveal their "repulsive" nature. Indeed, we show that the point processes can be characterized by an effective "hard-core" diameter that grows like the square root of d. The nearest-neighbor distribution functions for these point processes are also evaluated and rigorously bounded. Among other results, this analysis reveals that the probability of finding a large spherical cavity of radius r in dimension d behaves like a Poisson point process but in dimension d+1 for large r and finite d. We also show that as d increases, the point process behaves effectively like a sphere packing with a coverage fraction of space that is no denser than 1/2^d.